The cube is 997 is :

Options

A. `991026973`
B. `991029673`
C. `991029773`
D. `991097273`

Answer

To find the cube of 997, we can use the formula for the cube of a binomial, which is:

\[ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \]

In this case, we can express 997 as \(1000 - 3\). Therefore, we can set \(a = 1000\) and \(b = 3\).

### Step 1: Substitute into the formula
Using the values of \(a\) and \(b\):

\[ (1000 - 3)^3 \]

### Step 2: Apply the binomial expansion
Now we will apply the binomial expansion:

\[ = 1000^3 - 3 \times 1000^2 \times 3 + 3 \times 1000 \times 3^2 - 3^3 \]

### Step 3: Calculate each term
1. Calculate \(1000^3\):
\[ 1000^3 = 1000000000 \]

2. Calculate \(3 \times 1000^2 \times 3\):
\[ 3 \times 1000^2 \times 3 = 3 \times 1000000 \times 3 = 9000000 \]

3. Calculate \(3 \times 1000 \times 3^2\):
\[ 3 \times 1000 \times 9 = 27000 \]

4. Calculate \(3^3\):
\[ 3^3 = 27 \]

### Step 4: Combine all the terms
Now substitute all these values back into the equation:

\[ 997^3 = 1000000000 - 9000000 + 27000 - 27 \]

### Step 5: Simplify the expression
Now, let's simplify:

1. Start with \(1000000000 - 9000000\):
\[ 1000000000 - 9000000 = 991000000 \]

2. Now add \(27000\):
\[ 991000000 + 27000 = 991002700 \]

3. Finally, subtract \(27\):
\[ 991002700 - 27 = 991002673 \]

### Final Answer
Thus, the cube of 997 is:

\[ 997^3 = 991026973 \]